130 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 

 Differentiating now totally, we have 



dy "by dy 



dy r = -gfdt + ^dq, 



or on dividing through by dt, 



i dx r 



= ~dt 



3y r 



85) y;= 1nr 



We have now in each x',y',z', beside the linear function of q[, q' 2 , ... q^, 

 a term independent of the q tJ s, but which may be expressed 

 in terms of the coordinates q and t. On squaring there are accord- 

 ingly not only quadratic terms in the g"s, but also terms of the 

 first and zero orders. On forming the kinetic energy 



s1 



86) T = |VV 



we accordingly find that instead of being, as before, a homogeneous 

 function of the q"s, it contains not only quadratic terms, but also 

 terms linear in and others independent of the #"s. The effect of 

 these linear terms in the kinetic energy, whatever be their origin, 

 will be discussed in 50. 



40. Hamilton's Principle the most general dynamical 

 principle. We have seen in this chapter how by means of 

 Hamilton's Principle we may deduce the general equations of motion, 

 and from these the principle of Conservation of Energy. As 



